1.1. Lecture # 60: Introduction to wave mechanics (Contd.)

In this lecture, we proceed forward with the bottom boundary conditions also called as, BBC.

So, let us say the bottom is described as z = - h of x. So, if there is you know this is the riverbed or the seabed and this can be z here. So, if this is x and this is z. So, this depth z = - h f x because, we are considering the 0 at the free surface if we consider 0 at the free surface. So, here origin is located at still water level that is the surface of the water.

This section introduces the topic of bottom boundary conditions in wave mechanics. The bottom boundary condition (BBC) is essential for understanding how water waves interact with the ground below them, such as riverbeds or seabeds. The depth of the water at any point can be represented mathematically by the equation z = -h(x), where h is the height of the water at position x. The reference point for measuring depth (z) is chosen to be at the still water level, which is indicated as 0 on the z-axis. In other words, the depth increases negatively as you go down towards the seabed.

Think of the surface of a lake, which represents still water at a level you could walk on. If you were to imagine diving into the lake, the deeper you go, the more negative your depth would become below this water surface level. Hence, the further you dive, the more negative the numbers get, indicating how deep you are below this level.

Since from the boundary condition we see the bottom is fixed for you dot n is going to be 0, which we have seen in the previous lecture. So, we can ride the surface equation as so the equation was z = - h of x.

The concept of a fixed boundary at the bottom tells us that there is no vertical flow of water at this boundary (hence, u · n = 0). This means that water particles do not move through the seabed; they simply reflect back into the water. This fixed condition allows us to write a simple equation for the surface of the water, confirming z = -h(x), to show how the water level changes based on position x.

Imagine standing at the edge of a swimming pool. When you jump in, the water around you moves, but the bottom remains solid and unmoving. Just like that pool's bottom, in this context, water cannot move through the bottom - it bounces back, which is a fundamental behavior of water waves interacting with solid surfaces.

So, if we do dot product of u dot n what do what we get we get this multiplied by this multiplied by this. So, we get u dh dx + w is equal because here whatever is there 1 + the delta h delta x whole squared = 0.

Here, we discuss the relationship between velocity components as they relate to the bottom boundary. When considering the dot product of velocity (u) and the normal vector (n), the terms u·dh/dx and w play a crucial role in determining how velocities interact with the bottom surface. The equation simplifies to show that lateral (x) and vertical (z) velocities are interconnected via the slope of the water's surface (dh/dx), giving rise to important relationships between these components.

Think of cleaning a pool with a vacuum at the bottom. The vacuum must not only suck up dirt (moving water) but must correctly align to ensure it catches everything. Similarly, the velocity of the water moving with respect to the pool's bottom must work just right to ensure everything flows properly according to the slopes present.

Now, there is something called dynamic free surface boundary condition, free surface means, free surface of water is written as. So, you the surface is free it can distort if you recall from our open channel flow lectures.

The dynamic free surface boundary condition is essential for understanding how the water surface behaves under various circumstances. Unlike fixed boundaries, the surface can move or 'distort'. This behavior needs to be analyzed using specific equations, considering the impact of factors like wave heights and movement properties of the water at the free surface.

Think of a water balloon being gently shaken. As you shake it, the surface of the balloon wobbles and shifts; it does not remain perfectly still. This dynamic movement is analogous to how the surface of water in a wave can change position and shape depending on the forces acting on it, such as wind or the movement of the water underneath.

Therefore another boundary condition is required for any free surface or interface to prescribe the pressure distribution and the boundary.

At free surfaces, we cannot just rely on fixed conditions to explain how pressure acts. Because the surface can move, we need a different approach to establish how pressure varies across the free surface. This leads to the necessity of defining a dynamic boundary condition where we assume the pressure remains consistent across the surface, ensuring stability in our calculations and predictions.

Imagine blowing air over the surface of a glass of water. The air pressure above the water creates ripples and distorts the surface. Understanding how that pressure behaves over time and distance helps in predicting how the ripples will travel across the surface, helping us analyze wave motions effectively.

So, we see that we started from the boundary conditions, we went to the dynamic boundary condition where we saw the bottom boundary condition and kinematic free surface boundary condition.

In this concluding summary of the lecture, we reflect on the journey through various types of boundary conditions. The analysis began with examining fixed bottom boundaries and progressed towards exploring dynamic surface conditions. Through understanding these behaviors and conditions, we can make predictions and assess how water waves react in real-life scenarios, such as rivers, lakes, and oceans.

Picture a calm lake that suddenly experiences wind disturbances. By recognizing both the stable bottom and the dynamic surface behaviors, we can predict how the water will ripple and flow, similar to how understanding different boundary conditions helps engineers manage and design waterfront structures.

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